Local stabilization of discrete-time TS descriptor systems

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Local stabilization of discrete-time TS descriptor systems Zsófia Lendek a, *, Zoltán Nagy a , Jimmy Lauber b a b

Department of Automation, Technical University of Cluj-Napoca, Romania University of Valenciennes and Hainaut-Cambresis, LAMIH, Le Mont Houy, 59313 Valenciennes Cedex 9, France

a r t i c l e

i n f o

Keywords: Local stabilization TS models Descriptor form Robot arm

a b s t r a c t Descriptor models are naturally obtained from the Euler–Lagrange modeling approach to mechanical systems. Since the underlying system is nonlinear, global stabilization and/or tracking is possible only in a limited number of cases. Therefore, we develop conditions for local stabilization and tracking of discrete-time descriptor systems represented by Takagi–Sugeno fuzzy models, using both quadratic and nonquadratic Lyapunov functions. An estimate of the region of attraction is also obtained. The conditions are illustrated on a numerical example and in tracking control for a robot arm. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Mechanical systems (Lewis et al., 2004; Megahed, 1993; Spong et al., 2005) are frequently used in transportation, handling, assembly; security, surveillance, quality inspection, exploration can be done by mobile robots (Chan et al., 2013); and healthcare problems may be alleviated via bio-mechanic devices (Guelton et al., 2008; Nikdel et al., 2014). The dynamic models of such mechanical systems are generally determined using the Euler–Lagrange equations (Lewis et al., 2004; Spong et al., 2005), which give second-order vector differential equations. Once the Euler–Lagrange equations are obtained, the state-space representation is naturally in a descriptor form (Lewis et al., 2004; Luenberger, 1977; Chen et al., 2009), i.e., it has the mass matrix on the left-hand side. Since the model is nonlinear, nonlinear tools should be used for the analysis and design of such systems. In the last two decades, Takagi–Sugeno (TS) fuzzy models (Takagi and Sugeno, 1985) have attracted considerable interest in the automatic stability analysis and controller design of nonlinear systems. TS models represent the nonlinear system considered as a convex combination of local linear models blended together by nonlinear membership functions (MFs), which share the convex-sum property (Ohtake et al., 2001; Taniguchi et al., 2001). When using the sector nonlinearity approach (Ohtake et al., 2001) to obtain it, the TS model will be an exact representation of the system in the considered compact set of the statespace. A shortcoming of this approach is that the number of local models in the TS model is exponentially related to the number of nonlinearities in the original nonlinear model (Tanaka and Wang, 2001), leading to

increased computational costs. If descriptor models are represented in this classical form (via matrix inversion), this number grows very large. Instead, the goal of this paper is to develop conditions that ensure local asymptotic stability of discrete-time TS models directly in the descriptor form, i.e., we study the case when the considered equilibrium point is only locally stable. To our best knowledge, this is the first time that this topic has been addressed. We consider the problem of local stabilization and estimating a domain of attraction of the equilibrium point. We combine the tools existing in the discrete TS framework with the determination of a non-quadratic domain of attraction using an easy procedure that requires only the knowledge of the membership functions. We also extend this procedure to input-to-state stability (ISS). In order to obtain an exact TS representation of the original nonlinear model, we use the sector nonlinearity approach. A TS descriptor model (Taniguchi et al., 1999) generalizes the standard one and is obtained by applying the approach to both sides of the nonlinear differential equation. In this way, the nonlinear terms are separated and thus a smaller number of local models (Estrada-Manzo et al., 2015; Zhang and Feng, 2008; Chadli and Darouach, 2012) and a reduced number of LMI constraints (Guerra et al., 2007; Taniguchi et al., 2000) may be achieved. We develop conditions for local asymptotic stability and tracking. Since TS models are nonlinear, for their analysis and control synthesis the direct Lyapunov approach has been employed, using initially quadratic Lyapunov functions (Tanaka et al., 1998; Tanaka and Wang, 2001; Sala et al., 2005), then piecewise continuous Lyapunov functions (Johansson et al., 1999; Feng, 2004), and recently, nonquadratic Lyapunov functions (Guerra and Vermeiren, 2004; Kruszewski et al.,

* Corresponding author.

E-mail addresses: [email protected] (Zs. Lendek), [email protected] (Z. Nagy), [email protected] (J. Lauber). http://dx.doi.org/10.1016/j.engappai.2017.09.006 Received 15 January 2017; Received in revised form 7 August 2017; Accepted 4 September 2017 Available online xxxx 0952-1976/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.

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Engineering Applications of Artificial Intelligence (

2008; Mozelli et al., 2009), that have significantly improved existing results (Guerra and Vermeiren, 2004; Ding et al., 2006; Dong and Yang, 2009; Lee et al., 2011). The conditions for stability analysis or controller and observer design are developed in general in form of linear matrix inequalities (LMIs) that can be solved using available convex optimization methods (Boyd et al., 1994; Scherer and Weiland, 2005). Classic conditions, particularly in the discrete-time case, have been developed to establish global asymptotic stability of the (closed-loop) model, i.e., if the conditions are satisfied, then the model is globally asymptotically stable, but if they are unfeasible, no conclusions may be drawn. Thus, when the model is only locally stable or stabilizable, these conditions will not be satisfied. If the studied equilibrium point is only locally stable, the existing conditions become unusable, in the sense that the LMI conditions become unfeasible. In our previous works (Lendek and Lauber, 2016a, b) we have developed sufficient conditions for local stability and stabilization of classic discrete-time TS models. However, these results are not directly applicable to descriptor models, due to the descriptor matrix appearing on the left-hand side. In this paper we specifically address local stability and set point tracking for descriptor models. Next to developing local conditions, we also estimate the region of attraction of the considered equilibrium point. Results on this topic are scarce even for linear models. References (da Silva and Tarbouriech, 2001, 2006) consider linear discrete-time systems with actuator and/or state saturations and the stability analysis is performed using quadratic Lyapunov functions, while trying to find the maximum admissible quadratic domain of attraction. Others, considering the same problem of actuator saturation still using quadratic Lyapunov functions, proposed ways to design the domain of attraction based on convex set as polyhedrons (Hu et al., 2001) or as being saturation dependent (Cao and Lin, 2003). Here, we determine instead a non-quadratic domain of attraction based simply on the system’s dynamics. The paper is organized as follows. Section 2 presents the notations to be used throughout the paper and states the preliminaries. Section 3 concerns local stabilization of descriptor TS models. The conditions for local tracking are developed in Section 4. The conditions are discussed and illustrated on a numerical example and in tracking control of a robot arm in Section 5. Section 6 concludes the paper.

𝑟𝑒 ∑

𝑟 ∑

ℎ𝑖 (𝒛(𝑘))(𝐴𝑖 𝒙(𝑘) + 𝐵𝑖 𝒖(𝑘))

𝑖=1

where (𝐸𝑗 , 𝐴𝑖 , 𝐵𝑖 ) are local linear models, 𝒙 is the state vector, 𝒖 is the input vector, 𝒛 is the scheduling vector, 𝑟 and 𝑟𝑒 are the number of local linear models, 𝑣𝑗 (𝒛) and ℎ𝑖 (𝒛) are the membership functions that hold ∑𝑟𝑒 the convex sum property, i.e., 𝑣𝑗 (𝒛) ∈ [0, 1], 𝑗=1 𝑣𝑗 (𝒛) = 1, ℎ𝑖 (𝒛) ∈ [0, 1], ∑𝑟 𝑖=1 ℎ𝑖 (𝒛) = 1. TS models may be either an approximation of the nonlinear model (obtained e.g., by local linearization (Tanaka and Wang, 2001) or by substitution (Kiriakidis, 2007)) or an exact representation (when applying the sector nonlinearity approach Ohtake et al., 2001) in a compact set of the state-space. In this paper, the sector nonlinearity approach (Ohtake et al., 2001) is used, thus the resulting TS model will exactly represent the dynamic system in the considered compact set. The main idea of obtaining a fuzzy model using the sector nonlinearity approach (Ohtake et al., 2001) is as follows. Consider the nonlinear descriptor system 𝐸(𝒛(𝑘))𝒙(𝑘 + 1) = 𝒇 (𝒛(𝑘))𝒙(𝑘) + 𝒈(𝒛(𝑘))𝒖(𝑘)

(5)

𝒚(𝑘) = 𝐶𝒙(𝑘)

with 𝐸, 𝒇 and 𝒈 smooth nonlinear matrix functions, 𝒙 ∈ 𝑛 the state vector, 𝒖 ∈ 𝑛𝑢 the input vector, and 𝒚 ∈ 𝑛𝑦 the measurement vector, 𝒛 some vector function of 𝒙, 𝒚, all variables assumed to be bounded on a compact set . Let nl𝑗 (⋅) ∈ [nl𝑗 , nl𝑗 ], 𝑗 = 1, 2, … , 𝑝 be the set of bounded nonlinearities on the right-hand-side, i.e., components of either 𝒇 or 𝒈 𝑒 and nl𝑒𝑗 (⋅) ∈ [nl𝑒𝑗 , nl𝑗 ], 𝑗 = 1, 2, … , 𝑝𝑒 those on the left-hand-side, i.e., components of 𝐸. An exact TS fuzzy representation of (5) can be obtained by constructing first the weighting functions 𝑤𝑗0 (⋅) =

nl𝑗 − nl𝑗 (⋅)

𝑤𝑗1 (⋅) = 1 − 𝑤𝑗0 (⋅)

nl𝑗 − nl𝑗

𝑗 = 1, 2, … , 𝑝

𝑒

𝑤𝑗𝑒0 (⋅)

=

nl𝑗 − nl𝑒𝑗 (⋅) 𝑒

nl𝑗 − nl𝑒𝑗

𝑤𝑗𝑒1 (⋅) = 1 − 𝑤𝑗𝑒0 (⋅) 𝑗 = 1, 2, … , 𝑝𝑒

and defining the membership functions as ℎ𝑖 (𝒛) =

𝑝 ∏ 𝑗=1

𝑤𝑗𝑖 (𝒛𝑗 ) 𝑗

𝑣𝑖 (𝒛) =

𝑝𝑒 ∏ 𝑗=1

𝑤𝑗𝑒𝑖 (𝒛𝑗 ) 𝑗

(6)

with 𝑖 = 1, 2, … , 2𝑝 , 𝑖𝑗 ∈ {0, 1} and 𝑖 = 1, 2, … , 2𝑝𝑒 , 𝑖𝑗 ∈ {0, 1}, respectively. These membership functions are normal, i.e., ℎ𝑖 (𝒛) ≥ 0 ∑ ∑ 𝑟𝑒 and 𝑟𝑖=1 ℎ𝑖 (𝒛) = 1, 𝑟 = 2𝑝 , 𝑣𝑖 (𝒛) ≥ 0 and 𝑖=1 𝑣𝑖 (𝒛) = 1, 𝑟𝑒 = 2𝑝𝑒 , where 𝑟 and 𝑟𝑒 are the number of rules. The local matrices for each rule are obtained by substituting the corresponding upper or lower bounds of the nonlinearities. Using the membership functions defined in (6), an exact representation of (5) is given as:

(1)

where 𝒒 is the generalized position of the joints, 𝒒̇ is the velocity, 𝒒̈ is the ̇ is the Coriolis/centrifugal acceleration, 𝑀(𝒒) is the mass matrix, 𝐶𝑜 (𝒒, 𝒒) ̇ is the friction, 𝐺𝑟 (𝒒) is the gravity matrix, 𝝉 is the input matrix, 𝐹𝑟 (𝒒) vector. Note that the mass matrix is positive definite and thus in what follows we consider regular descriptor models. Rewriting (1) as ( )( ) ( ) ( ) 𝐼 0 𝒒̇ 𝒒̇ 0 = + 𝝉 (2) ̇ 𝒒̇ − 𝐹𝑟 (𝒒) ̇ − 𝐺𝑟 (𝒒)𝒒 0 𝑀(𝒒) 𝒒̈ −𝐶𝑜 (𝒒, 𝒒) 𝐼 ( ) 𝒒 a state-space representation of (2) may be obtained by defining 𝒙 = 𝒒̇ as the state vector and 𝒖 = 𝝉 as the input and can be written in the general form (Lewis et al., 2004)

𝑟𝑒 ∑ 𝑗=1

𝑣𝑗 (𝒛(𝑘))𝐸𝑗 𝒙(𝑘 + 1) =

𝑟 ∑

ℎ𝑖 (𝒛(𝑘))(𝐴𝑖 𝒙(𝑘) + 𝐵𝑖 𝒖(𝑘))

(7)

𝑖=1

where 𝒙 denotes the state vector, 𝑟 and 𝑟𝑒 are the number of rules on the right and left-hand side, respectively, 𝒛 is the scheduling vector, ℎ𝑖 , 𝑖 = 1, 2, … , 𝑟, 𝑣𝑗 , 𝑗 = 1, 2, … , 𝑟𝑒 are normalized membership functions, and 𝐴𝑖 , 𝐵𝑖 , 𝐸𝑗 , 𝑖 = 1, 2, … , 𝑟, 𝑗 = 1, 2, … , 𝑟𝑒 , are the local models.

(3)

Remark. Since we assume that 𝐸 is positive definite and thus it is invertible, by multiplying the nonlinear model with 𝐸 −1 one can obtain a classic state-space model and continue with a classic TS modeling. However, as shown by Estrada-Manzo et al. (2015), Guerra et al. (2007) and Taniguchi et al. (2000), keeping the descriptor form presents several advantages, among which having a smaller number of local models and obtaining a reduced number of LMI constraints. This is why we consider descriptor TS models.

with 𝐸(𝑥) being a positive definite matrix. Eq. (3) is in the general form of the so-called descriptor model (Luenberger, 1977). When discretizing (3), still a descriptor form is obtained: 𝐸(𝒙(𝑘))𝒙(𝑘 + 1) = 𝑓 (𝒙(𝑘), 𝒖(𝑘)).

𝑣𝑗 (𝒛(𝑘))𝐸𝑗 𝒙(𝑘 + 1) =

𝑗=1

The dynamics of mechanical systems is usually represented in a statespace form, which can be obtained from the Euler–Lagrange equations (Lewis et al., 2004):

𝐸(𝒙)𝒙̇ = 𝑓 (𝒙, 𝒖)

–

Takagi–Sugeno (TS) fuzzy models are convex combinations of local linear models, with the descriptor TS model (Taniguchi et al., 2000) having the form

2. Notation and preliminaries

̇ 𝒒̇ + 𝐹𝑟 (𝒒) ̇ + 𝐺𝑟 (𝒒)𝒒 = 𝝉 𝑀(𝒒)𝒒̈ + 𝐶𝑜 (𝒒, 𝒒)

)

(4)

In what follows, we will consider this discrete-time descriptor model. 2

Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.

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In this paper we develop sufficient conditions for the local stabilization of TS fuzzy models in descriptor form, i.e., of the form (7). Note that local results may also be obtained by simply linearizing the dynamics (4) around 𝒙 = 0 and computing a stabilizing state-feedback control law 𝒖 = −𝐹 𝒙. However, this linear control law in general will guarantee the stability of the closed-loop system only in a very small domain. By considering the TS descriptor representation (7) and developing local conditions, we aim to guarantee a much larger domain of attraction. In what follows, we use the following shorthand notations for generic matrices 𝑋: 𝑟𝑒

𝑋𝑣 ≡

∑

𝑋ℎ ≡

𝑣𝑖 (𝒛(𝑘))𝑋𝑖

𝑖=1 𝑟𝑒

𝑋𝑣ℎ ≡

𝑟 ∑

≡

Lemma 2 (Wang et al., 1996). The double convex sum (9) is negative, if 𝛤𝑖𝑖 < 0 𝑖, 𝑗 = 1, 2, … , 𝑟, 𝑖 < 𝑗.

(10)

ℎ𝑖 (𝒛(𝑘))𝑋𝑖

𝑖=1

𝑟 ∑∑

(

–

where 𝛤𝑖𝑗 , 𝑖, 𝑗 = 1, 2, … , 𝑟 are matrices of appropriate dimensions and the functions ℎ𝑖 (𝒛(𝑘)) satisfy the convex properties, i.e., ℎ𝑖 (𝒛(𝑘)) ∈ [0, 1] ∑ and 𝑟𝑖=1 ℎ𝑖 (𝒛(𝑘)) = 1, ∀𝒛(𝑘). While in the literature an abundance of sufficient conditions to ensure the negativity of the double sum above exist, for simplicity, in this paper we will use the following lemma.

𝛤𝑖𝑗 + 𝛤𝑗𝑖 < 0,

3. Local stabilization 𝑣𝑖 (𝒛(𝑘))ℎ𝑗 (𝒛(𝑘))𝑋𝑖𝑗

𝑖=1 𝑗=1 −1 𝑋𝑣ℎ

)

𝑟𝑒 𝑟 ∑ ∑

)−1 𝑣𝑖 (𝒛(𝑘))ℎ𝑗 (𝒛(𝑘))𝑋𝑖𝑗

Consider the descriptor TS model, repeated here for convenience: .

𝐸𝑣 𝒙(𝑘 + 1) = 𝐴ℎ 𝒙(𝑘) + 𝐵ℎ 𝒖(𝑘)

𝑖=1 𝑗=1

defined on a domain  including the origin. Our goal is to determine a control law of the form

Thus, (7) will be denoted as: 𝐸𝑣 𝒙(𝑘 + 1) = 𝐴ℎ 𝒙(𝑘) + 𝐵ℎ 𝒖(𝑘).

𝒖(𝑘) = −  −1 𝒙(𝑘)

Furthermore, 0 and 𝐼 denote the zero and identity matrices of appropriate dimensions, and the)term induced by symmetry ( a (∗)) denotes ( 𝐴 𝐵 𝐴 𝐵 in a matrix form, i.e., (∗) 𝐶 = 𝐵𝑇 𝐶 , and the symmetrical part of the expression on the left-hand side in inline form, i.e., 𝑋 + (∗) + 𝑌 = 𝑋 + 𝑋 𝑇 + 𝑌 . The subscripts ‘+’ (as in 𝐴ℎ+ , + ) and ‘−’ stand for the scheduling vector being evaluated at the next sample and at the previous sample, i.e., at 𝒛(𝑘 + 1) and at 𝒛(𝑘 − 1), respectively. We will also make use of the following results:

𝐸𝑣 𝒙(𝑘 + 1) = (𝐴ℎ − 𝐵ℎ   −1 )𝒙(𝑘)

Assumption 1. There exists a domain 𝑅 and a symmetric (possibly fuzzy) matrix expression  = 𝑇 so that ( )𝑇 ( ) 𝒙(𝑘) 𝒙(𝑘) (14)  ≥0 𝒙(𝑘 + 1) 𝒙(𝑘 + 1) holds ∀𝒙(𝑘) ∈ 𝑅 .

Proposition 1 (Congruence). Given a matrix 𝑃 = 𝑃 𝑇 and a full column rank matrix 𝑄 it holds that

Remark. Note that Assumption 1 expresses the domain 𝑅 as an implicit expression depending on the dynamics of the system, i.e., 𝑅 is given by } { ) )𝑇 ( |( 𝒙 𝒙 |  ≥ 0 𝑅 = 𝒙 ∈  | | 𝐸𝑣−1 (𝐴ℎ − 𝐵ℎ   −1 )𝒙 𝐸𝑣−1 (𝐴ℎ − 𝐵ℎ   −1 )𝒙 |

𝑃 > 0 ⇒ 𝑄𝑃 𝑄𝑇 > 0. Proposition 2. Let 𝐴 and 𝐵 be matrices of appropriate dimensions and ranks, with 𝐵 = 𝐵 𝑇 > 0. Then (𝐴 − 𝐵)𝑇 𝐵 −1 (𝐴 − 𝐵) ≥ 0 ⟺ 𝐴𝑇 𝐵 −1 𝐴 ≥ 𝐴 + 𝐴𝑇 − 𝐵.

For a more compact and easier notation, in developing the conditions we will use the form (14).

Proposition(3 (Schur Complement (Boyd et al., 1994)). Consider a matrix ) 𝑀 𝑀12 𝑀 = 𝑀 𝑇 = 𝑀11 , with 𝑀 𝑇 11 and 𝑀22 being square matrices. Then 𝑀 𝑀 <0⇔

22

𝑀11 < 0 ⇔ 𝑇 −1 𝑀22 − 𝑀12 𝑀11 𝑀12 < 0

{

(13)

has a locally asymptotically stable equilibrium point in 𝒙 = 0 and determine a region of attraction 𝑆 . For this, let us first assume the following

1. 𝒙𝑇 𝑄𝒙 < 0, 𝒙 ∈ {𝒙 ∈ R𝑛𝑥 , 𝒙 ≠ 0, 𝒙 = 0} 2. ∃ ∈ R𝑚×𝑛𝑥 such that 𝑄 +  + 𝑇 𝑇 < 0.

12

(12)

where  and  are (possibly fuzzy) controller gains such that the closedloop system,

Lemma 1 (Finsler’s Lemma (Skelton et al., 1998)). Consider a vector 𝒙 ∈ R𝑛𝑥 and two matrices 𝑄 = 𝑄𝑇 ∈ R𝑛𝑥 ×𝑛𝑥 and  ∈ R𝑚×𝑛𝑥 such that rank() < 𝑛𝑥 . The two following expressions are equivalent:

{

(11)

Assumption 1 above can always be satisfied, e.g., by choosing a constant matrix  = 𝑅 = 𝑅𝑇 > 0. The domain 𝑅 depends on the system being analyzed, and in the worst case 𝑅 = {0}.

𝑀22 < 0 −1 𝑇 𝑀11 − 𝑀12 𝑀22 𝑀12 < 0.

3.1. Local quadratic stabilization

Proposition 4 (S-Procedure). Consider matrices 𝐹𝑖 = 𝐹𝑖𝑇 ∈ R𝑛×𝑛 , 𝒙 ∈ R𝑛 , such that 𝒙𝑇 𝐹𝑖 𝒙 ≥ 0, 𝑖 = 1, … , 𝑝, and the quadratic inequality condition

We start with the following result. 𝒙𝑇 𝐹0 𝒙 > 0

(8) Theorem 1. Given a symmetric matrix expression  = 𝑇 such that Assumption 1 holds, the discrete-time nonlinear model (13) is locally asymptotically stable if there exist matrices 𝑃 = 𝑃 𝑇 > 0,  , , and scalar 𝜏 > 0 so that ( ) ( 𝑇 ) ( ) − −  𝑇 + 𝑃 (∗)  0  0 +𝜏  < 0 (15) 𝐴ℎ  − 𝐵ℎ  −𝐸𝑣 𝑃 − (∗) + 𝑃 0 𝑃 0 𝑃

𝒙 ≠ 0. A sufficient condition for (8) to hold is: there exist 𝜏𝑖 ≥ 0, 𝑖 = 1, … , 𝑝, such that 𝐹0 −

𝑝 ∑

𝜏𝑖 𝐹𝑖 > 0.

𝑖=1

Analysis and design for TS models often lead to double convex-sum negativity problems of the form 𝒙

𝑇

𝑟 ∑ 𝑟 ∑

ℎ𝑖 (𝒛(𝑘))ℎ𝑗 (𝒛(𝑘))𝛤𝑖𝑗 𝒙 < 0

Moreover, the region of attraction , i.e., the region from which all trajectories converge to zero, includes 𝑆 , where 𝑆 is the largest Lyapunov level set included in 𝑅 .

(9)

𝑖=1 𝑗=1

3

Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.

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Engineering Applications of Artificial Intelligence (

Proof. Consider the candidate Lyapunov function 𝑉 = 𝒙𝑇 (𝑘)𝑃 −1 𝒙(𝑘). The difference is

)

–

will depend on the controller gains and on the Lyapunov function. Moreover, since in the conditions (16), the matrix 𝑊 ‘‘compensates’’ for the difference in the ( Lyapunov ) functions, several possibilities can be chosen, 𝑊 0 such as 𝑊 = 01 −𝐼 , which establishes a direct relation between 𝒙(𝑘) and 𝒙(𝑘 + 1); a full 𝑊 , which will give a more complex relation between two consecutive samples and thus in principle a larger region, etc. In fact, as 𝑊 is used to determine a region where the difference in the Lyapunov function is negative, it should have a structure to match this difference, and ultimately to match the nonlinearities in the system model. Furthermore, since any trajectory that eventually gets in the domain 𝑆 is a stable one, the region of attraction may be increased by looking at those trajectories that do not start in 𝑅 , but arrive in several steps to 𝑆 . Let us now discuss the choice of the controller gains  and . Naturally, the simplest choice is  = 𝐹 ,  = 𝐻, 𝐹 and 𝐻 being matrices of appropriate dimensions. For such a choice, however, the controller is reduced to linear one, although (16) contains two sums (one in ℎ and one in 𝑣, thus relaxations cannot be applied). In order to be able to use relaxations such as Lemma 2,  and  should contain at least one sum in ℎ, i.e.,  = 𝐹ℎ and  = 𝐻ℎ . This means that (16) contains two sums in ℎ (for which the relaxations can be applied) and one in 𝑣. To increase the number of decision variables, and thus relax the conditions, a sum in 𝑣 can still be included in  and , leading to  = 𝐹ℎ𝑣 and  = 𝐻ℎ𝑣 , respectively. Note that this choice does not lead to an increase in the number of sums. Finally, continuing the line of thought above, for the general case, one may choose  = 𝐹ℎℎ…ℎ𝑣…𝑣 and  = 𝐻ℎℎ…ℎ𝑣…𝑣 . By increasing the number of sums – and, equivalently, the number of decision variables – , the controller becomes more general and the LMIs less conservative. However, at the same time, the computational complexity increases, eventually rendering the conditions computationally intractable in practice. It should be noted that depending on the control problem considered other choices (e.g., using a constant 𝐻 and  = 𝐹ℎ𝑣 ) may also be suitable. Moreover, choosing a constant 𝐻 = 𝑃 recovers classic controllers from the literature.

𝛥𝑉 = 𝒙𝑇 (𝑘 + 1)𝑃 −1 𝒙(𝑘 + 1) − 𝒙𝑇 (𝑘)𝑃 −1 𝒙(𝑘) ( )𝑇 ( −1 )( ) 𝒙(𝑘) −𝑃 0 𝒙(𝑘) = . −1 𝒙(𝑘 + 1) 𝒙(𝑘 + 1) 0 𝑃 In the domain 𝑅 Assumption 1 holds, thus, using Proposition 4, we have 𝛥𝑉 < 0 if there exists 𝜏 > 0 so that ( )𝑇 ( −1 )( ) 𝒙(𝑘) −𝑃 0 𝒙(𝑘) −1 𝒙(𝑘 + 1) 𝒙(𝑘 + 1) 0 𝑃 ( )𝑇 ( ) 𝒙(𝑘) 𝒙(𝑘) +𝜏  ≤ 0. 𝒙(𝑘 + 1) 𝒙(𝑘 + 1) Furthermore, the dynamics (11) can be written as ( ) ( ) 𝒙(𝑘) = 0. 𝐴ℎ − 𝐵ℎ   −1 −𝐸𝑣 𝒙(𝑘 + 1) Using Lemma 1, we have 𝛥𝑉 < 0 if there exist  so that ( −1 ) ( ) −𝑃 0  𝐴ℎ − 𝐵ℎ   −1 −𝐸𝑣 + (∗) + −1 + 𝜏 < 0. 0 𝑃 ( ) 0 Choosing  = 𝑃 −1 leads to (

) −𝑃 −1 (∗) −1 −1 −1 −1 + 𝜏 < 0. 𝑃 𝐴ℎ − 𝑃 𝐵ℎ   −𝑃 𝐸𝑣 + (∗) + 𝑃 ( 𝑇 )  0 Congruence with 0 and applying Proposition 2 gives directly 𝑃 conditions (15) and concludes the proof. Moreover, the region of attraction includes 𝑆 , where 𝑆 is the largest Lyapunov level set included in 𝑅 . □ −1

Before discussing the choice of the structure of  and , let us now consider the case when  is to be determined. Our aim is to effectively find the domain 𝑆 where the system (13) is locally asymptotically stable. For this, similarly to the result in Lendek and Lauber (2016b), we consider  as a decision variable, and, with a slight abuse of notation, 𝜏 will be denoted simply by . Thus, the following result can be formulated.

3.2. Local non-quadratic stabilization Theorem 2. The discrete-time descriptor model (13) is locally asymptotically stable in the domain 𝑆 if there exist matrices 𝑃 = 𝑃 𝑇 > 0,  , , 𝑖 = 1, 2, … , 𝑟 and 𝑊 = 𝑊 𝑇 so that ( ) − −  𝑇 + 𝑃 (∗) +𝑊 <0 (16) 𝐴ℎ  − 𝐵ℎ  −𝐸𝑣 𝑃 − (∗) + 𝑃 ⋂ where 𝑆 is the largest Lyapunov level set included in 𝑅  and 𝑅 is given by {

In what follows, we will consider local nonquadratic stabilization of the descriptor model (13). For this we will use the nonquadratic Lyapunov function 𝑉 = 𝒙𝑇 (𝑘) −1 𝒙(𝑘), together with the constraint ( )𝑇 ( ) 𝒙(𝑘) 𝒙(𝑘)  𝒙(𝑘 + 1) between consecutive samples. Then, following the 𝒙(𝑘 + 1) lines of Theorem 2, we have the result: Theorem 3. The discrete-time nonlinear model (13) is locally asymptotically stable in the domain 𝑆 , if there exist ,  , , and , so that ( ) − 𝑇 −  +  (∗) + <0 (17) 𝐴ℎ  − 𝐵ℎ  −𝐸𝑣 + + (∗) + +

𝒙(𝑘) ∈  |

𝑅 = (

𝒙(𝑘) 𝒙(𝑘 + 1)

)𝑇 (

} ) ( −1 )( )  −𝑇 0  0 𝒙(𝑘) 𝑊 ≥ 0 𝒙(𝑘 + 1) 0 𝑃 −1 0 𝑃 −1

where 𝑅 is given by { 𝒙(𝑘) ∈  |

𝑅 = Proof. The proof follows the same lines as that of Theorem 1 and denoting ) ( −1 ) ( −𝑇  0  0 = −1 𝑊 −1 . □ 0 𝑃 0 𝑃

} )𝑇 ( −𝑇 ) ( −1 )( )  0  0 𝒙(𝑘)  ≥0 𝒙(𝑘 + 1) 0 +−1 0 +−1 ⋂ and 𝑆 is the largest Lyapunov level set included in 𝑅 .

Remark. Classical ( ) results in the literature are usually based on the 0 choice  = 𝑀 , in Lemma 1. In this particular case, we chose

Proof. The proof follows the same lines as that of Theorem 2 and is therefore omitted. □

(

𝒙(𝑘) 𝒙(𝑘 + 1)

𝑀 = 𝑃 −1 .

Once the sums (membership functions) to be used in  , , , etc., are chosen, sufficient LMI conditions can easily be derived for the above conditions. However, in order to efficiently apply relaxations such as

As can be seen, the domain 𝑅 and consequently the domain where local asymptotic stability of the closed-loop system will be established 4

Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.

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(

)𝑇

Theorem 4. The discrete-time nonlinear model (20) is locally ISS with respect to the exogenous input 𝒚𝑟 in the domain 𝑆 , if there exist ,  , , and , so that ( ) − 𝑇 −  +  (∗) + <0 (21) 𝑒 𝑒 𝑒 𝐴ℎ  − 𝐵ℎ  −𝐸𝑣 + + (∗) + + where 𝑅 is given by { 𝒙𝑒 (𝑘) ∈  |

𝑅 = (

The domain 𝑅 is obtained based on , 𝐻ℎ and 𝑃𝑣 . While even a constant matrix 𝑊 may lead to a nonconvex set, in order to maximize the degrees of freedom in , one can choose  = 𝑊ℎℎ𝑣𝑣𝑣− , i.e., include all the indices that appear on the left-hand side. Then, the following sufficient LMI conditions may be formulated:

𝒙𝑒 (𝑘) 𝑒 𝒙 (𝑘 + 1)

)𝑇 (

} ) ( −1 )( 𝑒 )  −𝑇 0  0 𝒙 (𝑘)  ≥ 0 𝒙𝑒 (𝑘 + 1) 0 +−1 0 +−1

and 𝑆 is the largest Lyapunov level set included in 𝑅

The dynamics (20) can be written as ( 𝑒 𝐴ℎ − 𝐵ℎ𝑒   −1

𝒙𝑒 (𝑘) ⎞ )⎛ 𝑒 ⎜ 𝐷 𝒙 (𝑘 + 1)⎟ = 0 ⎜ ⎟ ⎝ 𝒚𝑟 ⎠

−𝐸𝑣𝑒

𝒙(𝑘) ∈  | )𝑇 (

Then, we have

} ) ( −1 )( ) 𝐻ℎ−𝑇 0 𝐻ℎ 0 𝒙(𝑘)  ≥ 0 . 𝒙(𝑘 + 1) 0 𝑃𝑣−1 0 𝑃𝑣−1

( 𝛥𝑉 =

)𝑇 (

𝒙𝑒 (𝑘) 𝒙𝑒 (𝑘 + 1)

− −1

0

0

+−1

)(

𝒙𝑒 (𝑘)

)

𝒙𝑒 (𝑘 + 1)

𝑇

4. Local set point tracking control Consider now the discrete-time TS descriptor system (11) 𝐸𝑣 𝒙(𝑘 + 1) = 𝐴ℎ 𝒙(𝑘) + 𝐵ℎ 𝒖(𝑘)

=

(19)

𝒚(𝑘) = 𝐶𝒙(𝑘)

) ( ) ( 𝐵 𝐸 , 𝐵ℎ𝑒 = 0ℎ , 𝐸𝑣𝑒 = 0𝑣

0 𝐼

=

=

− 𝐵ℎ𝑒   −1 )𝒙𝑒 (𝑘) + 𝐷𝒚𝑟 .

𝒙𝑒 (𝑘 + 1)

𝒙𝑒 (𝑘)

)𝑇 (

𝑒

𝒙 (𝑘 + 1)

− −1 +−1 (𝐴𝑒ℎ

−

)(

(∗)

𝐵ℎ𝑒   −1 )

−+−1 𝐸𝑣𝑒

+ (∗) +

+−1

𝒙𝑒 (𝑘)

)

𝑒

𝒙 (𝑘 + 1)

and 𝛿 ≥ 0 and 𝛼 ≥ 0 are bounding constants, i.e., ‖+−1 (𝐴ℎ −𝐵ℎ   −1 )‖ ≤ 𝛿 and ‖+−1 𝐷‖ ≤ 𝛼. Let us now consider (𝒙𝑒 ). Assuming that there exists  and a domain 𝑅 such that ( 𝑒 )𝑇 ( 𝑒 ) 𝒙 (𝑘) 𝒙 (𝑘)  ≥0 𝒙𝑒 (𝑘 + 1) 𝒙𝑒 (𝑘 + 1)

) ,

and we consider a control law 𝒖(𝑘) = −  −1 𝒙𝑒 (𝑘). The closed-loop system is (𝐴𝑒ℎ

⎞ ⎛ 𝒙𝑒 (𝑘) ⎞ ) ⎟ ⎟⎜ 𝐷 + (∗)⎟ ⎜𝒙𝑒 (𝑘 + 1)⎟ ⎟ ⎟⎜ 𝒚 𝑟 ⎠ ⎠⎝ )( ) 𝑒 (∗) 𝒙 (𝑘)

𝒙𝑒 (𝑘 + 1) +−1 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 ) −+−1 𝐸𝑣𝑒 + (∗) + +−1 ( 𝑒 ) −1 + 2𝒚𝑟 + (𝐴ℎ − 𝐵ℎ𝑒   −1 )𝒙𝑒 (𝑘) + 𝐷𝒚𝑟

(𝒙𝑒 ) (

𝐸𝑣𝑒 𝒙𝑒 (𝑘 + 1) = 𝐴𝑒ℎ 𝒙𝑒 (𝑘) + 𝐵ℎ𝑒 𝒖(𝑘) + 𝐷𝒚𝑟

𝐸𝑣𝑒 𝒙𝑒 (𝑘 + 1)

−𝐸𝑣𝑒

where

𝐸𝑣 𝒙(𝑘 + 1) = 𝐴ℎ 𝒙(𝑘) + 𝐵ℎ 𝒖(𝑘)

0 𝐼

⎛ 𝒙𝑒 (𝑘) ⎞ ⎛⎛ 0 ⎞ ⎟( ⎜ ⎟ ⎜⎜ + ⎜𝒙𝑒 (𝑘 + 1)⎟ ⎜⎜+−1 ⎟ 𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 ⎜ 𝒚 ⎟ ⎜⎜ 0 ⎟ 𝑟 ⎠ ⎝ ⎠ ⎝⎝ ( )𝑇( 𝑒 𝒙 (𝑘) − −1

≤ (𝒙𝑒 ) + 2𝛿‖𝒚𝑟 ‖‖𝒙𝑒 (𝑘)‖ + 2𝛼‖𝒚𝑟 ‖2

defined on a domain  including the origin, where 𝒚 denotes the output. Our goal is to determine a control law, such that 𝒚 tracks a desired reference signal 𝒚𝑟 . For this, we will use the auxiliary state variable 𝒙𝐼 (𝑘) – corresponding to an integral term – with the dynamics given by 𝒙𝐼 (𝑘 + 1) = 𝒙𝐼 (𝑘) − 𝐶𝒙(𝑘) + 𝒚𝑟 . The extended state dynamics are 𝒙𝐼 (𝑘 + 1) = 𝒙𝐼 (𝑘) − 𝐶𝒙(𝑘) + 𝒚𝑟 . ( ) ( 𝒙(𝑘) 𝐴 Denoting 𝒙𝑒 = 𝒙𝐼 (𝑘) , 𝐴𝑒 = −𝐶ℎ ( ) 0 𝐷 = 𝐼 we have

.

𝛥𝑉 = (𝒙𝑒 )𝑇 (𝑘 + 1)+−1 𝒙𝑒 (𝑘 + 1) − (𝒙𝑒 )𝑇 (𝑘) −1 𝒙𝑒 (𝑘) ( 𝑒 )𝑇 ( −1 )( 𝑒 ) 𝒙 (𝑘) − 0 𝒙 (𝑘) = . 𝑒 𝑒 −1 𝒙 (𝑘 + 1) 𝒙 (𝑘 + 1) 0 +

Moreover, the region of attraction includes 𝑆 , where 𝑆 is the largest Lyapunov level set included in 𝑅 , where 𝑅 is defined as {

𝒙(𝑘) 𝒙(𝑘 + 1)

⋂

Proof. Consider the candidate Lyapunov function 𝑉 = (𝒙𝑒 )𝑇 (𝑘) −1 𝒙𝑒 (𝑘). The difference in 𝑉 is

Corollary 1. The discrete-time descriptor model (13) is locally asymptoti𝑇 cally stable if there exist matrices 𝑃𝑘 = 𝑃𝑘𝑇 > 0, 𝐻𝑖 , 𝐹𝑖 , 𝑊𝑖𝑗𝑘𝑙𝑚 = 𝑊𝑖𝑗𝑘𝑙𝑚 , 𝑖, 𝑗 = 1, 2, … , 𝑟𝑒 , 𝑘, 𝑙, 𝑚 = 1, 2, … , 𝑟, so that (10) hold, with ( ) −𝐻𝑖 − 𝐻𝑖𝑇 + 𝑃𝑚 (∗) 𝛤𝑖,𝑗,𝑘,𝑙,𝑚 = + 𝑊𝑖𝑗𝑘𝑙𝑚 𝐴𝑗 𝐻𝑖 − 𝐵𝑗 𝐹𝑖 −𝐸𝑘 𝑃𝑙 + (∗) + 𝑃𝑙

(

–

( 𝑒 ) 𝒙 (𝑘)  𝒙𝑒 (𝑘 + 1) between consecutive samples. Then, following the lines of Theorem 2, we have the result: 𝒙𝑒 (𝑘) 𝒙𝑒 (𝑘 + 1)

Lemma 2 to reduce the computational complexity, the sums used in the Lyapunov function and in the controller gains should be suitably chosen. A similar line of choices of the sums used in the control gains and in the Lyapunov function as in the case of quadratic stabilization can be discussed also here. Since, due to the nonquadratic Lyapunov function, + appears in the conditions, the choice of delayed Lyapunov matrices, such as those by Guerra et al. (2012) may be warranted. When using fuzzy expression in both the gains and the Lyapunov function, the least number of sums that allows for the use of relaxations both in ℎ and 𝑣 is 5:  = 𝐻ℎ ,  = 𝐹ℎ ,  = 𝑃𝑣− , where 𝑣− denotes the membership functions 𝑣 evaluated at the previous sampling instant. This choice leads to ( ) −𝐻ℎ − 𝐻ℎ𝑇 + 𝑃𝑣− (∗) +  < 0. (18) 𝐴ℎ 𝐻ℎ − 𝐵ℎ 𝐹ℎ −𝐸𝑣 𝑃𝑣 + (∗) + 𝑃𝑣

𝑅 =

)

(𝒙𝑒 ) < 0, if ( − −1 −1 𝑒 + (𝐴ℎ − 𝐵ℎ𝑒   −1 ) ( 𝑇  Congruence with 0

(20)

It is well-known (Lendek et al., 2010) that if the closed-loop system (20) with 𝒚𝑟 = 0 is globally uniformly asymptotically stable, then it is also input-to-state stable (ISS) with respect to the exogenous input 𝒚𝑟 . In what follows, we determine conditions for the local tracking and determine a region where tracking is possible. Similarly to the stabilization problem, consider the nonquadratic Lyapunov function 𝑉 = 𝒙𝑒𝑇 (𝑘) −1 𝒙𝑒 (𝑘), together with the constraint

( − 𝑇 −  +  𝐴𝑒ℎ  − 𝐵ℎ𝑒 

or ( − 𝑇 −  +  𝐴𝑒ℎ  − 𝐵ℎ𝑒 

−+−1 𝐸𝑣𝑒 0 +

)

(∗) + (∗) + +−1

+  < 0.

and applying Proposition 2 gives

(∗) −𝐸𝑣𝑒 + + (∗) + +

−𝐸𝑣𝑒 +

)

)

+

(∗) + (∗) + +

( 𝑇  0

) ( 0   + 0

0 +

) < 0 (22)

) +  < 0.

5

Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.

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Engineering Applications of Artificial Intelligence (

Furthermore in 𝑅 where (22) holds, ∃𝜆 > 0 so that (𝒙𝑒 ) < −𝜆‖𝒙𝑒 (𝑘)‖2 . Consequently, in this domain,

and by the completion of squares, 2𝛿‖𝒚𝑟 ‖‖𝒙𝑒 (𝑘)‖ ≤ 𝜃1 ‖𝒙𝑒 (𝑘)‖2 +𝛿 2 𝜃‖𝒚𝑟 ‖2 , ∀𝜃 > 0, thus

minimize 𝛼, 𝛿, maximize 𝜆 subject to (23), (24), (25)

1 𝛥𝑉 ≤ −𝜆‖𝒙𝑒 (𝑘)‖2 + ‖𝒙𝑒 (𝑘)‖2 + 𝛿 2 𝜃‖𝒚𝑟 ‖2 + 2𝛼‖𝒚𝑟 ‖2 . 𝜃 1 𝜆

and denoting 𝑐1 = 𝜆 −

1 𝜃

–

Corollary 2. The discrete-time nonlinear model (20) is locally ISS with respect to the exogenous input 𝒚𝑟 in the domain 𝑆 , if there exist ,  , , and , so that condition (21) holds. Furthermore, the ultimate bound on the states can be minimized by solving

𝛥𝑉 ≤ −𝜆‖𝒙𝑒 (𝑘)‖2 + 2𝛿‖𝒚𝑟 ‖‖𝒙𝑒 (𝑘)‖ + 2𝛼‖𝒚𝑟 ‖2

Choosing 𝜃 >

)

together with (21).

> 0 and 𝑐2 = 𝛿 2 𝜃 + 2𝛼, we have

𝛥𝑉 ≤ −𝑐1 ‖𝒙𝑒 (𝑘)‖2 + 𝑐2 ‖𝒚𝑟 ‖2 . Furthermore, consider 𝜏 ∈ (0, 1). Then,

5. Examples

𝛥𝑉 ≤ −(1 − 𝜏)𝑐1 ‖𝒙𝑒 (𝑘)‖2 − 𝜏𝑐1 ‖𝒙𝑒 (𝑘)‖2 + 𝑐2 ‖𝒚𝑟 ‖2 𝑐 ≤ −(1 − 𝜏)𝑐1 ‖𝒙𝑒 (𝑘)‖2 ∀‖𝒙𝑒 ‖2 ≥ 2 ‖𝒚𝑟 ‖2 𝜏𝑐1

In this section we illustrate the proposed conditions first on a numerical example and then on a 2DOF robot arm.

i.e., the closed-loop system (20) is ISS with respect to the exogenous 𝑐 input 𝒚𝑟 , with an ultimate bound given by 𝜏𝑐2 . □ 1

While the bounding constants 𝛼 and 𝛿 do not affect the feasibility of the developed LMI conditions, they do affect the bound above and how closely the systems output 𝒚 will track the reference signal 𝒚𝑟 . Thus, in what follows, we develop LMI conditions for the minimization of this bound. Recall that ‖+−1 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 )‖ ≤ 𝛿 and ‖+−1 𝐷‖ ≤ 𝛼 and consider first ‖+−1 𝐷‖ ≤ 𝛼. This is satisfied if

5.1. Numerical example

Consider the discrete-time descriptor TS model: 2 ∑

𝐷𝑇 +−1 +−1 𝐷 ≤ 𝛼 2 𝐼

𝑣𝑖 (𝒛(𝑘))𝐸𝑖 𝒙(𝑘 + 1) =

with (𝑥1 (𝑘), 𝑥2 (𝑘)) ∈ R2 , 𝒛 = 𝒙,

By the Schur complement, we have ) ( 2 𝛼 𝐼 𝐷𝑇 +−1 ≥ 0. −1 + 𝐷 𝐼

1 − sin(𝑥1 ) 𝑣2 = 1 − 𝑣1 2 1 − sin(𝑥1 ) 1 − sin(𝑥2 ) 1 − sin(𝑥1 ) 1 + sin(𝑥2 ) ℎ1 = ℎ2 = 2 2 2 2 1 + sin(𝑥1 ) 1 − sin(𝑥2 ) 1 + sin(𝑥1 ) 1 + sin(𝑥2 ) ℎ3 = ℎ4 = 2 2 2 2 𝑣1 =

Congruence with diag(𝐼 + ) gives ( 2 ) 𝛼 𝐼 𝐷𝑇 ≥0 𝐷 + +

(

and using Proposition 2 on + + we obtain ( 2 ) 𝛼 𝐼 𝐷𝑇 ≥ 0. 𝐷 2+ − 𝐼

𝐸1 = (

(23) 𝐴1 =

Consider now ‖+−1 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 )‖ ≤ 𝛿. This is satisfied, if

( 𝐴3 =

(+−1 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 ))𝑇 +−1 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 ) ≤ 𝛿 2 𝐼. Using the Schur complement and congruence with diag( 𝑇 + ) gives ( )  𝑇 𝛿2  (∗) ≥0 𝐴𝑒ℎ  − 𝐵ℎ𝑒  + + and using Proposition 2 on both  𝑇 𝛿 2  and + + leads to ) ( 1 (∗) 𝑇 +  − 𝐼 ≥ 0. 𝛿2 𝐴𝑒ℎ  − 𝐵ℎ𝑒  2+ − 𝐼

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𝑐

(∗) ⎞ ( 0 ⎟+  ⎟ 0 1 0 − 𝐼 ⎟⎠ 𝜆 This result can be summarized as follows. (∗) + (∗) + +

) 0 < 0. 0

0 1

)

−0.55 −0.85 0.48 −0.6

( 𝐸2 = ) −0.25 0.66 ) −0.15 −0.43

( 𝐴2 = ( 𝐴4 =

−0.21 1.55 ⎞ ⎛ 1.36 𝑃 = ⎜−0.21 2.34 −1.63⎟ ⎜ ⎟ −1.63 5.95 ⎠ ⎝ 1.55 ( ) −1 𝐹1,1 𝐻 = −1.06 0.68 0.25 ( ) 𝐹2,1 𝐻 −1 = −1.39 −0.05 0.25 ( ) 𝐹3,1 𝐻 −1 = −0.89 −0.42 0.25 ( ) −1 𝐹4,1 𝐻 = 1.28 −1.30 0.25

which, after applying the Schur complement results in −𝐸𝑣𝑒 +

1 0

1 1

1 3

)

−0.85 −1.20 −0.08 1.53

𝐵=

( ) 0 1

) −0.12 −0.06 ) −0.60 . −1.35

Our goal is to determine a controller that is able to track the reference signals presented in Figs. 1(a) and 1(b). Using classical conditions for the global stabilization of this system, the LMIs are unfeasible. Thus, we consider a local approach. For the easier graphical representation, we 𝑒 𝑒𝑇 𝑒 consider ( )a common quadratic Lyapunov function 𝑉 (𝒙 ) = 𝒙 (𝑘)𝑃 𝒙 (𝑘), 𝒙 𝑒 𝑇 𝒙 = 𝒙𝐼 with 𝑃 = 𝑃 > 0 and the following choices:  = 𝑃 ,  = 𝐹ℎ𝑣 , ( ) 𝑊 0 𝑊 = 01 −𝐼 . Note that with the choice of  = 𝑃 the controller is reduced to a classic PDC controller. The following results have been obtained:

Furthermore, since the bound 𝜏𝑐2 depends on 𝑐1 , which in turn 1 depends on 𝜆, 𝜆 should be maximized. Then, (𝒙𝑒 ) ≤ −𝜆‖𝒙𝑒 (𝑘)‖, if ( ) − −1 + 𝜆 (∗) −1 𝑒 𝑒 −1 −1 𝑒 −1 +  < 0. + (𝐴ℎ − 𝐵ℎ   ) −+ 𝐸𝑣 + (∗) + + ( 𝑇 )  0 Similarly to the proof of Theorem 4, congruence with 0 and + applying Proposition 2 gives ( ) − 𝑇 −  + 𝜆 𝑇  +  (∗) + <0 𝐴𝑒ℎ  − 𝐵ℎ𝑒  −𝐸𝑣𝑒 + + (∗) + + ⎛− 𝑇 −  +  ⎜ 𝐴𝑒  − 𝐵 𝑒  ℎ ⎜ ℎ ⎜ 𝑇 ⎝

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ℎ𝑖 (𝒛(𝑘))(𝐴𝑖 𝒙(𝑘) + 𝐵𝒖(𝑘))

𝑖=1

𝑖=1

𝛼 2 𝐼 − 𝐷𝑇 +−1 +−1 𝐷 ≥ 0.

4 ∑

0 0 ⎞ ⎛0.28 𝑊1 = ⎜ 0 0.28 0 ⎟ ⎜ ⎟ 0 0.28⎠ ⎝ 0 ( ) −1 𝐹1,2 𝐻 = −0.48 1.03 0.46 ( ) 𝐹2,2 𝐻 −1 = −0.34 0.11 0.45 ( ) 𝐹3,2 𝐻 −1 = −1.79 −0.21 0.45 ( ) −1 𝐹4,2 𝐻 = 1.20 −0.39 0.46 .

As can be seen in the tracking results illustrated in Figs. 1(c) and 1(d), the tracking is indeed only possible locally: for instance, the reference 𝑦𝑟 = −2 (see Fig. 1(b)) cannot be tracked – at least with this choice of the control law and the Lyapunov functions –, although for 𝑦𝑟 = −0.2, 𝑥1 converges to the reference.

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6

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The domain in this case is determined by { ( 𝑒 )𝑇 ( −1 )( 𝒙 (𝑘) 𝑃 0 𝑊1 𝑅 = 𝒙𝑒 (𝑘) ∈  | 𝑒 −1 𝒙 (𝑘 + 1) 0 0 𝑃 } ( −1 )( 𝑒 ) 𝑃 0 𝒙 (𝑘) × ≥0 𝒙𝑒 (𝑘 + 1) 0 𝑃 −1 ⎧ ⎪ = ⎨𝒙𝑒 (𝑘) ∈  | 𝒙𝑒 (𝑘)𝑇 ⎪ ⎩

⎛ 0.35 ⎜−0.07 ⎜ ⎝−0.12

⎛−1.26 + 𝒙𝑒 (𝑘 + 1)𝑇 ⎜ 0.25 ⎜ ⎝ 0.45

0.25 −0.34 −0.19

⎧ ⎪ = ⎨𝒙𝑒 ∈  | 𝒙𝑒𝑇 ⎪ ⎩

⎛ 0.35 ⎜−0.07 ⎜ ⎝−0.12

−0.07 0.1 0.05

0 −𝐼

𝐿1 [m] 𝐿2 [m] 𝑀1 [kg] 𝑀2 [kg] g [m/s2 ] 𝐼1𝑥 [kgm2 ] 𝐼1𝑦 [kgm2 ] 𝐼1𝑧 [kgm2 ] 𝐼2𝑥 [kgm2 ] 𝐼2𝑦 [kgm2 ] 𝐼2𝑧 [kgm2 ] 𝑏1 [−] 𝑏2 [−]

−0.12⎞ 0.05 ⎟ 𝒙𝑒 (𝑘) ⎟ 0.06 ⎠

0.25 −0.34 −0.19

0.095 0.1 0.0.095 0.37 9.81 2.27 ⋅ 10−2 8.33 ⋅ 10−5 1.81 ⋅ 10−5 8.33 ⋅ 10−5 2.27 ⋅ 10−2 7.07 ⋅ 10−5 0.24 0.16

Length first–second joint Length second joint end-effector Mass first joint Mass second joint Gravitational acceleration Moment of inertia Moment of inertia Moment of inertia Moment of inertia Moment of inertia Moment of inertia Friction coefficient, first joint Friction coefficient, second joint

𝐿2 cos(𝑥2 ) 2 ⎛ 2 ) ⎜𝐼1𝑥 + 𝐼2𝑧 + cos(𝑥2 ) (𝐼2𝑥 − 𝐼2𝑧 ) + 𝑀2 (𝐿1 + 2 𝑀(𝒙) = ⎜ ⎜ 0 ⎝

−0.12⎞ 0.05 ⎟ 𝒙𝑒 ⎟ 0.06 ⎠

⎛−1.26 + 𝒙𝑒𝑇 (𝐴𝑒ℎ − 𝐵ℎ𝑒   −1 )𝑇 (𝐸𝑣𝑒𝑇 )−1 ⎜ 0.25 ⎜ ⎝ 0.45

–

Table 1 Margin settings.

)

⎫ 0.45 ⎞ ⎪ −0.19⎟ 𝒙𝑒 (𝑘 + 1) ≥ 0⎬ ⎟ ⎪ −0.22⎠ ⎭

−0.07 0.1 0.05

)

0 𝑀2 𝐿22

𝑀2 𝐿22 ⎛ + 𝐼2𝑥 − 𝐼2𝑧 ) + 𝐿1 𝐿2 𝑀2 sin(𝑥2 )) + 𝑏1 ⎜−𝑥4 (sin(2𝑥2 )( 4 𝐷(𝒙) = ⎜ sin(2𝑥2 ) 𝐿2 𝐿 𝑀 𝐿 sin(𝑥2 ) ⎜ (𝐼 − 𝐼 )+( 2 2 1 ) 2𝑥 2𝑧 + ⎝ 2 4 2

⎫ 0.45 ⎞ ⎪ −0.19⎟ (∗)𝒙𝑒 ≥ 0⎬ . ⎟ ⎪ −0.22⎠ ⎭

4 ⎞ 0⎟ ⎟. 𝑏2 ⎟⎠

⎞ ⎟ ⎟ + 𝐼2𝑦 ⎟⎠

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( ) −3𝜋 3𝜋 The state variables are bounded as 𝑞1 , 𝑞2 ∈ and 𝑞̇ 1 , 𝑞̇ 2 ∈ , 4 4 ( ) −3, 3 . Using a simple forward Euler discretization with sampling

Since the Lyapunov function depends on 𝑥1 , 𝑥2 and 𝑥𝐼 , its projection on the 𝑥1 −𝑥2 plane, i.e., the level sets for 𝑥𝐼 = 0, and those points (denoted by ‘o’ ) which are in 𝑅 are presented in Fig. 1(e). The points that are in 𝑅 for 𝑥1 ∈ [−1, 1], 𝑥2 , 𝑥𝐼 ∈ [−2, 2] are graphically represented in Fig. 1(f). Two issues need to be noted here:

time 𝑇𝑠 = 0.045 and the sector nonlinearity approach, an equivalent TS descriptor model is:

∙ Although not the case for asymptotic stabilization, in case of ISS, the obtained domain also depends on the auxiliary integral variable 𝒙𝐼 . Since this variable directly depends on the exogenous input 𝒚𝑟 , it naturally follows that stability will only be ensured for some reference signals. ∙ The domain of attraction is actually determined by an implicit equation of the state (stabilization) and of 𝒚𝑟 (in case of ISS). Depending on the nonlinearities in the membership functions, the choice of the sums in the controller gains, and the (possibly delayed) Lyapunov matrix, analytically solving this equation may not be possible.

2 ∑

𝑣𝑖 (𝒛(𝑘))𝐸𝑖 𝒙(𝑘 + 1) =

4 ∑

ℎ𝑖 (𝒛(𝑘))(𝐴𝑖 𝒙(𝑘) + 𝐵𝒖(𝑘))

𝑖=1

𝑖=1

with the matrices

Remark. For comparison purposes we also computed a linear statefeedback control law 𝒖 (= −𝐹 𝒙 for the linearized ) model, and obtained the controller gain 𝐹 = −0.29 −0.35 −0.05 . However, this control cannot even stabilize the original nonlinear system from the initial point ( )𝑇 𝒙 = 0.1 0.1 , not to mention tracking a reference. This clearly illustrates the advantages of designing a local nonlinear controller. 5.2. Tracking for a robot arm In what follows, we consider a tracking problem for the 2DOF robot arm shown in Fig. 2(a). The schematic representation of this arm is shown in Fig. 2(b). Using the Euler–Lagrange modeling approach, the dynamic model of this arm can be obtained as

⎛1 ⎜ 0 𝐸1 = ⎜ ⎜0 ⎜0 ⎝

0 1 0 0

0 0 0.0242 0

0 ⎞ ⎟ 0 ⎟ 0 ⎟ 0.0236⎟⎠

⎛1 ⎜ 0 𝐸2 = ⎜ ⎜0 ⎜0 ⎝

0 1 0 0

0 0 0.0307 0

0 ⎞ ⎟ 0 ⎟ 0 ⎟ 0.0236⎟⎠

⎛1 ⎜ 0 𝐴1 = ⎜ ⎜0 ⎜0 ⎝

0 1 0 0

0.045 0 0.0132 −0.003

0 ⎞ ⎟ 0.045 ⎟ 0 ⎟ 0.0164⎟⎠

⎛1 ⎜ 0 𝐴2 = ⎜ ⎜0 ⎜0 ⎝

0 1 0 0

0.045 0 0.0132 0.003

0 ⎞ ⎟ 0.045 ⎟ 0 ⎟ 0.0164⎟⎠

⎛1 ⎜ 0 𝐴3 = ⎜ ⎜0 ⎜0 ⎝

0 1 0 0

0.045 0 0.02 −0.003

0 ⎞ ⎟ 0.045 ⎟ 0 ⎟ 0.0164⎟⎠

0 1 0 0

0.045 0 0.02 0.003

𝐵 = 0.045𝐵𝑐

𝐶=

⎛1 ⎜ 0 𝐴4 = ⎜ ⎜0 ⎜0 ⎝ ) 0 . 0

(

1 0

0 1

0 0

0 ⎞ ⎟ 0.045 ⎟ 0 ⎟ 0.0164⎟⎠

Similarly to the numerical example in the previous section, our goal

(27) ̇ 𝒒̇ + 𝐼𝜏 𝑀(𝒒)𝒒̈ = − 𝐷(𝒒, 𝒒) ( )𝑇 ( )𝑇 where 𝒒 = 𝑞1 𝑞2 are the angles of the two joints, 𝒒̇ = 𝑞̇ 1 𝑞̇ 2 are ( )𝑇 the angular velocities, 𝜏 = 𝜏1 𝜏2 are the torques. 𝑀 represents the mass matrix and 𝐷 contains the Coriolis, centrifugal and friction forces. The parameters of the system have been experimentally identified and are presented in Table 1. The model (27) is naturally in descriptor form. With the notation ( )𝑇 𝒙 = 𝑞1 𝑞2 𝑞̇ 1 𝑞̇ 2 , we have ( ) ( ) ( ) 𝐼 0 0 𝐼 0 𝐸𝑐 (𝒙) = 𝐴𝑐 (𝒙) = 𝐵𝑐 = 0 𝑀(𝒙) 0 −𝐷(𝒙) 𝐼

is to determine a controller such that the angles 𝑥1 and 𝑥2 track a given reference signal. Using classical conditions for the global stabilization of this system, the LMIs for the descriptor form are unfeasible. Thus, we consider a local approach. We consider a common quadratic Lyapunov function 𝑉 (𝒙𝑒 ) = 𝒙𝑒𝑇 (𝑘)𝑃 𝒙𝑒 (𝑘), with 𝑃 𝑇 > 0 and the following ( 𝑃 = ) 𝑊1 0 choices:  = 𝑃 ,  = 𝐹ℎ𝑣 , 𝑊 = 0 . The obtained Lyapunov −𝐼 matrix, 𝑊 and some of the gains are: 7

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(a) Reference signal for Example 5.1.

)

–

(b) Reference signal for Example 5.1.

(c) Results for the reference in Fig. 1(a).

(d) Results for the reference in Fig. 1(b).

(e) Lyapunov level sets corresponding to 𝑥𝐼 = 0 and points that are in 𝑅 .

(f) The domain 𝑅 .

Fig. 1. Results for Example 5.1.

⎛ 0.13 ⎜ 0 ⎜ −0.05 𝑃 =⎜ ⎜ 0 ⎜ 0.68 ⎜ ⎝ 0

0 0.13 −0.00 −0.05 0 0.68

−0.05 0 1.27 0 −0.5 0

0 −0.05 0 1.26 0 −0.50

0.68 0 −0.5 0 9.92 0

0 ⎞ 0.68 ⎟ ⎟ 0 ⎟ −0.50⎟ 0 ⎟⎟ 9.92 ⎠

The tracking results are presented in Fig. 3. Note that the domain in which the tracking controller is locally asymptotically stable is determined by { } ( 𝑒 )𝑇 ( −1 )( 𝑒 ) 𝒙 (𝑘) 𝑃 𝑊1 𝑃 −1 0 𝒙 (𝑘) 𝑅 = 𝒙(𝑘) ∈  | ≥ 0 𝒙𝑒 (𝑘 + 1) 𝒙𝑒 (𝑘 + 1) 0 𝑃 −2

0 −0.1 0 0.04 0 ⎞ ⎛−0.10 ⎜ 0 −0.1 0 −0.1 0 0.4 ⎟ ⎜ ⎟ −0.1 0 1.23 0 −0.18 0 ⎟ 𝑊1 = ⎜ ⎜ 0 −0.1 0 1.22 0 −0.18⎟ ⎜ 0.04 0 −0.18 0 1.95 0 ⎟⎟ ⎜ ⎝ 0 0.04 0 −0.18 0 1.95 ⎠ ( ) 4.18 0 0.47 0 −0.22 0 𝐹1,1 𝐻 −1 = 0 4.2 −0.006 0.54 0 −0.22 ( ) 4.11 0 0.62 0 −0.21 0 𝐹4,2 𝐻 −1 = . 0 4.2 0.006 0.54 0 −0.22.

thus it will also depend on the reference signal to be tracked. Finally, the experimental results obtained for the 2DOF robot arm are shown in Fig. 4. Since the angular velocities for this robot arm are actually not measured, a linear observer has been used to estimate them. The observer has been computed for the linearized model by placing the poles of the estimation error dynamics at 0.6, 0.65, 0.7, and 0.75. The 0 ⎞ ⎛ 0.3 0 0.34⎟ observer gain is 𝐿 = ⎜0.83 . The controller uses the measured angles 0 ⎟ ⎜ ⎝ 0 0.23⎠ and the estimated angular velocities. 8

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(a) A 2DOF robot arm.

)

–

(b) Schematic representation of a 2DOF robot arm.

Fig. 2. A 2DOF robot arm.

(a) Tracking results for 𝑥1 .

(b) Tracking results for 𝑥2 .

Fig. 3. Tracking results for a robot arm.

(a) Tracking results for 𝑥1 .

(b) Tracking results for 𝑥2 .

Fig. 4. Experimental results for the robot arm.

9

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6. Conclusions

)

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In this paper, we have developed conditions for the local asymptotic stability and input-to-state-stability of discrete-time descriptor TS models. The conditions have been formulated as linear matrix inequality conditions that are easy to solve. An estimate of the domain of attraction has also been determined based on the Lyapunov function used and on the system matrices. The application of the proposed conditions has been illustrated on a numerical example and experimentally on a 2DOF robot arm. In our future work, we will extend the conditions for general delayed Lyapunov functions and controller gains. Acknowledgments This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project number PN-II-RU-TE-2014-4-0942, contract number 88/01.10.2015. References Boyd, S., El Ghaoui, L., Féron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in System and Control Theory. In: Stud. Appl. Math., Society for Industrial and Applied Mathematics, Philadelphia, PA, USA. Cao, Y., Lin, Z., 2003. Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function. Automatica 39 (7), 1235–1241. Chadli, M., Darouach, M., 2012. Novel bounded real lemma for discrete-time descriptor systems: Application to control design. Automatica 48 (2), 449–453. Chan, R.P.M., Stol, K.A., Halkyard, C.R., 2013. Review of modelling and control of twowheeled robots. Annu. Rev. Control 37 (1), 89–103. Chen, Y.-J., Wang, W.-J., Chang, C.-L., 2009. Guaranteed cost control for an overhead crane with practical constraints: Fuzzy descriptor system approach. Eng. Appl. Artif. Intell. 22, 639–645. da Silva, J., Tarbouriech, S., 2001. Local stabilization of discrete-time linear systems with saturating controls: an LMI-based approach. IEEE Trans. Automat. Control 46 (1), 119–124. da Silva, J., Tarbouriech, S., 2006. Anti-windup design with guaranteed regions of stability for discrete-time linear systems. Systems Control Lett. 55 (3), 184–192. Ding, B., Sun, H., Yang, P., 2006. Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-Sugeno’s form. Automatica (ISSN: 00051098) 42 (3), 503–508. Dong, J., Yang, G., 2009. Dynamic output feedback 𝐻∞ control synthesis for discrete-time T-S fuzzy systems via switching fuzzy controllers. Fuzzy Sets and Systems 160 (19), 482–499. Estrada-Manzo, V., Lendek, Zs., Guerra, T.M., Pudlo, P., 2015. Controller design for discrete-time descriptor models: a systematic LMI approach. IEEE Trans. Fuzzy Syst. 23 (5), 1608–1621. Feng, G., 2004. Stability analysis of discrete-time fuzzy dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 12 (1), 22–28. Guelton, K., Delprat, S., Guerra, T.-M., 2008. An alternative to inverse dynamics joint torques estimation in human stance based on a Takagi-Sugeno unknown-inputs observer in the descriptor form. Control Eng. Pract. 16 (12), 1414–1426. Guerra, T.M., Bernal, M., Kruszewski, A., Afroun, M., 2007. A way to improve results for the stabilization of continuous-time fuzzy descriptor models. In: Proceedings of the 46th IEEE Conference on Decision and Control, pp. 5960–5964. Guerra, T.M., Kerkeni, H., Lauber, J., Vermeiren, L., 2012. An efficient Lyapunov function for discrete TS models: observer design. IEEE Trans. Fuzzy Syst. 20 (1), 187–192. http://dx.doi.org/10.1109/TFUZZ.2011.2165545.

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Please cite this article in press as: Lendek Zs., et al., Local stabilization of discrete-time TS descriptor systems. Engineering Applications of Artificial Intelligence (2017), http://dx.doi.org/10.1016/j.engappai.2017.09.006.